Question

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U0 are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

Column I	Column II
(A) $U_1\left(x\right)=\frac{U_0}{2}{[1-{\left(\frac{x}{a}\right)}^2]}^2$	(P) The force acting on the particle is zero at x = a.
(B) $U_2\left(x\right)=\frac{U_0}{2}{\left(\frac{x}{a}\right)}^2$	(Q) The force acting on the particle is zero at x = 0.
(C) $U_3\left(x\right)=\frac{U_0}{2}{\left(\frac{x}{a}\right)}^2\exp [-{\left(\frac{x}{a}\right)}^2]$	(R) The force acting on the particle is zero at x = – a.
(D) $U_4\left(x\right)=\frac{U_0}{2}[\frac{x}{a}-\frac{1}{3}{\left(\frac{x}{a}\right)}^3]$	(S) The particle experiences an attractive force towards x = 0 in the region $|x|<a.$
	(T) The particle with total energy $\frac{U_0}{4}$can oscillate about the point x = – a.

• A. $\left(A\right)\to (P,Q,T);\left(B\right)\to (Q,T)$
  $\left(C\right)\to (P,Q.R,S)$
  $\left(D\right)\to (P,R,T)$
• B. $\left(A\right)\to (P,Q,R,T);\left(B\right)\to (Q,S)$
  $\left(C\right)\to (P,Q.R,S)$
  $\left(D\right)\to (P,R,T)$
• C. $\left(A\right)\to (P,Q,R,T);\left(B\right)\to (Q,S)$
  $\left(C\right)\to (P,Q.R,S)$
  $\left(D\right)\to (P,R,T)$
• D. $\left(A\right)\to (P,Q,R,T);\left(B\right)\to (Q,S)$
  $\left(C\right)\to (P,Q,S,T)$
  $\left(D\right)\to (P,R,S)$

Answer 1

The correct option is C $\left(A\right)\to (P,Q,R,T);\left(B\right)\to (Q,S)$

$\left(C\right)\to (P,Q.R,S)$
$\left(D\right)\to (P,R,T)$

For $A$: $\overrightarrow{F}=-\frac{dU}{dx}\hat{i}=-\frac{U_0}{2}2(1-{\left(\frac{x}{a}\right)}^2)\times [-2\left(\frac{x}{a}\right)\times \frac{1}{a}]\hat{i}=2U_0[1-{\left(\frac{x}{a}\right)}^2]\left[\frac{x}{a^2}\right]\hat{i}$

$Ifx=0\Rightarrow \overrightarrow{F}=\frac{U_0}{2}[2\left(1\right)\times 0]=\overrightarrow{0},U=\frac{U_0}{2}$

$Ifx=a\Rightarrow \overrightarrow{F}=\overrightarrow{0},andU=0$

$Ifx=-a\Rightarrow \overrightarrow{F}=\overrightarrow{0},andU=0$
so, $A\to P,Q,R,T$

For $B$: $\overrightarrow{F}=-\frac{U_0}{2}\times 2\left(\frac{x}{a}\right)\times \frac{1}{a}\hat{i}=-\frac{U_{0}x}{a^2}\hat{i}$

$Ifx=0\Rightarrow \overrightarrow{F}=0,andU=0$

$Ifx=a\Rightarrow \overrightarrow{F}=-\frac{U_0}{2}\hat{i}andU=\frac{U_0}{2}$

$Ifx=-a\Rightarrow \overrightarrow{F}=+\frac{U_0}{2}\hat{i}andU=\frac{U_0}{2}$
so, $B\to Q,S$

For $C$:
$U_3\left(x\right)=\frac{U_0}{2}{\left(\frac{x}{a}\right)}^2\exp [-{\left(\frac{x}{a}\right)}^2]$
,
for force, $F=U_0{e}^{\frac{-x^2}{a^2}}\frac{x}{a}[\frac{x^2}{a^2}-1]$
when, $x=0;U_0=0F=0;x=0,x=a,x=-a$
so, $C\to P,Q,R,S$

For $D$:
$U_4\left(x\right)=\frac{U_0}{2}[\frac{x}{a}-\frac{1}{3}{\left(\frac{x}{a}\right)}^3]$

on differentiating the above equation.
$\frac{dU}{dt}=F=\frac{U_0}{2a}[1-\frac{x^2}{a^2}]$
So, for $U=0;x=0,x=+\sqrt{3}a,x=-\sqrt{3}a$
$F=0;x=+a,x=-a$
so, $D\to P,R,T$

Hence,
$A\to P,Q,R,T$
$B\to Q,S$
$C\to P,Q,R,S$
$D\to P,R,T$